I describe an example of a finitely generated infinite simple group. is amenable and simple, but not finitely generated).

1. Construction

Start with a Cantor space , a minimal self homeomorphism of . The topological full group is the group of all homeos of such that for all , there exists such that .

Theorem 1 (Matui 2006) If is a minimal subshift (there exist uncounably many of them), the commutator subgroup of is simple and finitely generated.

2. Proof of amenability

We view as a group of piecewise-translations of , i.e. such that is bounded. Indeed, orbits are dense, so it suffices to look at element s of acting on one single orbit, i.e. on . The group of piecewise-translations contains (van Douwen).

2.1. Our strategy

1. We let act in a clever manner on finite subsets of , which is a group.

2. The action of on has locally finite (and thus amenable) stabilizers.

3. There exists a -invariant measure on . This suffices to prove that is amenable.

2.2. Action of

We embed into the semi-direct product as follows: is mapped to . This yields the following action of on : .

2.3. Amenability of stabilizers

Now is minimal action of on has ubiquitous pattern property, which means that for every finite set of elements , on a large interval of for a suitable picked nearby any point of .

We show that this implies that stabilizers are amenable.

To get the invariant mean on , it suffices to construct a sequence of -almost invariant functions in . We set

These are a modification of functions used by Kechris and Tsankov for the standard -action on . They used characteristic functions of sets ; on takes value 1 more often than 0.